Simulating the Local Behaviour of Small Pedestrian Groups Ioannis Karamouzas∗

Mark Overmars∗

Center for Advanced Gaming and Simulation, Utrecht University

(a)

(b)

(c)

(d)

Figure 1: Example test-case scenarios: (a) A group has to adapt its formation to pass through a doorway, (b) Group interactions in a confined environment, (c) A faster group overtakes a slower moving group, (d) A group of three agents walking through a narrow corridor.

Abstract

to three members, such as couples and friends going for shopping or families walking together [Coleman and James 1961; Moussa¨ıd et al. 2010].

Recent advancements in local methods have significantly improved the collision avoidance behaviour of virtual characters. However, existing methods fail to take into account that in real-life pedestrians tend to walk in small groups, consisting mainly of pairs or triples of individuals. We present a novel approach to simulate the walking behaviour of such small groups. Our model describes how group members interact with each other, with other groups and individuals. We highlight the potential of our method through a wide range of test-case scenarios. A number of metrics are also proposed to quantitatively evaluate the quality of our proposed model.

Over the past few decades, a number of approaches have been proposed for simulating group motions. Nevertheless, most of these methods focus on large groups of virtual characters or on how such groups are formed and not on the dynamics of small groups and on how group members interact and behave within a crowd. In addition, approaches based on flocking rules, as well as leader-follower models are mainly applicable to simulate the collective behaviour of large herds or flocks. The requirements, though, for guiding the motion of small pedestrian groups are different; friends or couples prefer to walk next to each other, rather than following a leader.

CR Categories: I.2.9 [Robotics]: Kinematics and dynamics; I.2.1 [Applications and Expert Systems]: Games

1

More recently, several approaches have tried to address the issue of realistic behaviour of small groups of virtual humans based on empirical observations of real crowds. These methods are able to capture the macroscopic behaviour of small groups, generating formations similar to the ones observed in real pedestrian groups. The problem, though, is that, in the resulting simulations, the group members lack anticipation and prediction, which sometimes leads to unrealistic microscopic behaviours, such as oscillations or backward motions. Additionally, due to the fact that such methods are mainly based on rules or social forces, they often require careful parameter tuning to generate desired group movements.

Introduction

Virtual worlds are ubiquitous in video games, training applications and animation films. Such worlds, to become more lively and appealing, are populated by a large number of characters. Typically, these characters should be able to navigate through the virtual environment in a human-like manner, avoiding collisions with other characters and the static part of the environment. As a result, a realistic and physically correct simulation of virtual humans has become a necessity for interactive worlds and games.

Contributions. In this paper, we present a novel approach to simulate the walking behaviour of small groups of characters. The focus of our work is on the local behaviour of such groups, that is on how group members interact with each other, with other groups and individual agents. Our proposed model is elaborated from recent empirical studies regarding the spatial organization of pedestrian groups [Moussa¨ıd et al. 2010] and is complementary to existing methods for solving interactions between virtual characters.

Current state-of-the-art techniques for real-time crowd simulation rely on agent-based solutions. In these systems, the global motion of each agent is typically governed by a higher-level path planning approach, whereas local interactions are resolved using behavioural rules. While recent advancements in local methods have significantly improved the collision avoidance behaviour of virtual characters, the majority of existing studies treat crowds as a collection of individual agents. However, in real-life, most of the pedestrians do not walk alone, but in small groups consisting mainly of two

In contrast to prior approaches, we use the velocity space to plan the avoidance maneuvers of each group, striving to maintain a configuration that facilitates the social interactions between the group members. The final motion of each individual member is then computed by an underlying agent-based algorithm. We demonstrate the potential and flexibility of our approach against a wide range of test case scenarios, as can be seen in Figure 1 as well as in the companion video. In all of our simulations, the groups exhibited convincing behaviour, smoothly avoiding collisions with other groups and individuals. We show that even in challenging scenarios, the groups safely navigate toward their goals by dynamically adapting

∗ e-mail:{ioannis,markov}@cs.uu.nl

Copyright © 2010 by the Association for Computing Machinery, Inc. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions Dept, ACM Inc., fax +1 (212) 869-0481 or e-mail [email protected]. VRST 2010, Hong Kong, November 22 – 24, 2010. © 2010 ACM 978-1-4503-0441-2/10/0010 $10.00

183

their formations, just like groups of pedestrians do in real-life. A quantitative evaluation of our model is also presented that allows us to objectively assess the steering quality of our model and verify the emergence of empirically observed walking patterns. Organization. The rest of the paper is organized as follows. Section 2 provides an overview of prior work related to our research. In Section 3, we define the basic problem and outline our proposed solution. A detailed explanation of our approach is provided in Sections 4 and 5, whereas experiments to show its usability are presented in Section 6. In Section 7, we provide a qualitative comparison with earlier solutions and highlight the limitations of our model. Finally, some conclusions and plans for further research are discussed in Section 8.

2

(a) Line-Abreast

(b) V-like

(c) River-like

Figure 2: Group formations according to [Moussa¨ıd et al. 2010] Prior work in graphics and animation community has also focused on the synthesis of realistic group motions. Kwon et al. [2008] proposed a technique that allows the user to interactively edit existing group motions, whereas Takahashi et al. [2009] used a spectralbased approach to control group formations in applications like mass performances and tactical sports. Recently, example-based approaches have also been used to construct group behaviour models from motion capture data or from videos of real-crowds [Lee et al. 2007; Lerner et al. 2007]. However, these approaches are too computationally expensive for real-time interactive applications and are commonly used for offline crowd simulations.

Related Work

The most common way to model the locomotion of human crowds is with agent-based methods, in which each agent plans individually its own actions. In such approaches, global path planning and local collision avoidance are typically decoupled. We refer the reader to [LaValle 2006] for an extensive literature on global navigation techniques. Regarding the microscopic (local) behaviour of individual agents, numerous models have been proposed, including force-based approaches [Helbing et al. 2000; Pelechano et al. 2007; Karamouzas et al. 2009], behavioural models [Funge et al. 1999; Shao and Terzopoulos 2007] and variants of velocity-based methods [Paris et al. 2007; van den Berg et al. 2008; Guy et al. 2009; Pettr´e et al. 2009].

3

Definitions and Background

Our approach is directly inspired by the recent work of Moussa¨ıd et al. [2010] and focuses on the local behaviour of small pedestrian groups. In [Moussa¨ıd et al. 2010], empirical data of pedestrian crowds were collected using video recordings of urban areas. The analysis of the corresponding data has shown that the majority of the pedestrians walk in small groups consisting of up to three members. In addition, regarding the spatial organization of pedestrian groups, three distinct formations can be observed, as shown in Figure 2. Typically, group members tend to walk next to each other forming a line perpendicular to the walking direction (line abreast formation). Such a formation allows the pedestrians to easily communicate with each other while advancing toward their goal. At moderate crowd densities, the group space is reduced and a “Vlike” formation emerges, facilitating the social communication between the group members. Finally, at high densities, safety prevails over social interactions and group members choose to walk behind each other, which results in a “river-like formation” (leaderfollower model). Note that for groups of two pedestrians, the Vshape formation is replaced by a more compact abreast formation, in which the security distance between the group members is significantly reduced.

Agent-based modeling has also been used to simulate the behaviour of groups of virtual entities. The work of Reynolds on boids has been influential in this field [1987]. Reynolds used simple local rules to create visually compelling flocks of birds and schools of fishes. Later he extended his model to include additional steering behaviours for autonomous agents [Reynolds 1999]. Since his original work, many interesting models have been introduced for controlling group motions. Loscos et al. [2003] presented a leader-follower model in which the leader decides about the motion of the entire group and the rest of the group members follow. Musse and Thalmann [1997] defined a rule-based model that allows virtual humans to switch groups based on sociological factors. Brogan and Hodgins [1997] accounted for motion dynamics while simulating groups of humanoid characters. Braun et al. [2003] expanded Helbing’s social force model and used attractive forces to form groups of pedestrians, whereas Qiu et al. [2008] used behavioural approaches to model different group structures in pedestrian crowds. More recently, Peters and Ennis [2009] proposed a rule-based model to simulate plausible behaviours of small groups consisting of up to four individuals based on observations from real crowds. Similarly, Moussa¨ıd et al. [2010] have conducted a series of studies to gain more insight into the organization of pedestrian groups in urban environments and introduced a force model that accounts for social interactions among people walking in groups.

3.1

Problem Formulation

In our problem setting, we are given a geometric description of the virtual environment, either 2D or 3D, in which small groups of agents must move. In case of a 3D environment, we assume that the agents are moving on a plane or terrain and represented as discs, resulting in a 2D motion planning problem. Based on the aforementioned empirical observations, we simulate groups of up to three agents; we assume that agents in larger groups tend to form smaller subgroups consisting of pairs or triples of individuals as also noted in pedestrian literature [Coleman and James 1961]. Given a group Gi having size 1 < N ≤ 3, we denote the position, radius and the velocity of an agent Aij belonging to Gi as xij , rij and vij respectively, where j ∈ [1, N ]. During each simulation cycle, we also define the centroid Ci of the group, i.e. the average position of all the group members. Similarly, we determine the current velocity Vi of the group as the average velocity of its members. Finally, each group has a desired velocity Vides indicating the preferred speed and desired direction of motion of its members.

At the global level, Kamphuis and Overmars [2004] developed a method for planning the motion of coherent groups using the concept of path planning inside corridors, while Bayazit et al. [2003] combined flocking techniques with probabilistic roadmaps to guide the flock members toward their goals. In the robotics community, centralized planners have also been exploited to compute the simultaneous motion of multiple units [Li and Chou 2003]. Nevertheless, the running time of such approaches grows exponentially with the number of robots. An alternative approach is based on continuum dynamics which attempts to directly guide the global behaviour of large homogeneous groups using continuous density fields [Hughes 2003; Treuille et al. 2006].

184

For each group member

For each group of characters

Compute desired velocity from group’s Vnew and F new

Compute new velocity Vnew and formation F new

Local collision avoidance

Figure 4: Example of interpolation between a current and a riverlike formation to derive a candidate formation.

Update position

as an input to a local collision avoidance model which returns the new velocity for the group agent.

Figure 3: Schematic overview of our proposed framework.

4

Based on the empirically observed walking patterns of pedestrian groups, we assume that each group Gi has k desired formations Fk , k = 2 + N , which in order of decreasing preference are the abreast, V-like and river-like formations (see Figure 2). Note that the group has as many river formations as the number N of its members. Each Fk Fk k formation Fk is characterized by the tuple (pF i , oij ), where pi Fk represents the reference point of the formation and oij describes the relative position of each group member with respect to the reference point. In case of the abreast or the V-shape formation, the reference point is represented by the centroid of the group, whereas in the river-like formation, the reference point is designated by the position of the leader agent.

This section elaborates on the first phase of our approach. The goal here is to determine at each simulation step the new velocity Vnew and formation F new of each group entity Gi , so that the group can safely navigate toward its target, ensuring at the same time that its members will stay as coherent as possible. We propose a velocitybased model consisting of four steps: Step 1. In the first step, we determine the set of admissible formations AF for the group. Ideally, the group members prefer to walk next to each other in an abreast formation. However, in crowded or rather confined environments this can lead to deadlocks. Imagine for example two pairs of agents walking down a narrow corridor from opposite directions. If there is not enough room for both groups to fit through the corridor, the agents will get stuck. To resolve these challenging scenarios, the group should be able to dynamically adapt its formation, just like groups of pedestrians do in real-life. As a result, we consider a number of alternative formations by linearly interpolating between the current formation of the group and its k desired formations. Since Equation (1) provides the desired position for each group agent Aij in the template formation Fk , our linear interpolation scheme over the interval s ∈ [0, 1] can be formulated as follows:

To be able to express each desired formation Fk in a coordinateinvariant way and use it throughout the simulation, we express the F relative positions oijk in a local coordinate system centered at the formation’s reference point and oriented toward the group’s desired Vdes

direction of motion n = kVides k . Consequently, at any time step of i the simulation, we can determine the desired location of the agent that is currently positioned at xij as: xijk = pi k + oijk [0]n + oijk [1]n⊥ F

F

F

F

(1)

The desired formations of the groups are part of the problem description and are given in advance (e.g. by the level designer). In our simulations, some noise was also introduced to allow for variation in the three template formations. For convenience, we also assume that initially the group members are placed at their abreast formation. The problem can then be characterized as follows. The group agents Aij , j ∈ [1, N ], need to reach a specified goal area without colliding with the environment and with each other, as well as with other individuals or groups that may be present in the virtual world. In addition, the agents must walk close together as a coherent group, striving to maintain a formation that supports the social communication among them. We assume that the goal is reached when all the agents of the group are within the goal area.

3.2

Avoidance Maneuvers for the Group

F

xij (s)

=

(1 − s)xij + sxijk ,

pi (s)

=

k (1 − s)Ci + spF i .

(2)

An example of such an interpolation is depicted in Figure 4, where the current formation of the group combined with a river-like formation (s = 0.5) results in a diagonal stripe pattern. Step 2. In the second step, we define the personal space of each candidate formation F cand belonging to the set AF . Just like an individual is protected from unwanted social and physical contact by its personal space, a portable territory is also formed around a group formation indicating the area that others should not invade. In our model, this personal area is conservatively approximated as an axis aligned bounding box (see Figure 5). The aabb can be efficand ciently computed using the local coordinates oF of the candiij date formation. We first consider the coordinate frame centered at the reference point of the candidate formation and oriented toward the group’s desired velocity. Then, using a min-max representation of the personal space, the minimum and maximum coordinate values along each local axis are computed as follows:

Overall Approach

We propose a two-phase approach to solve the aforementioned motion planning problem (Figure 3). In our setting, we assume that at every cycle of the simulation the desired velocity of each group is provided by some higher level path planning approach. Then, in the first step of our algorithm, we determine an “optimal” avoidance maneuver for each group of agents. We formulate this as a discrete optimization problem of finding an optimal new formation and velocity for the entire group. In the second step of our approach, we use the computed solution velocity and formation to determine the desired velocity of each group member. This velocity is then given

N

N

j=1 N

j=1 N

j=1

j=1

xmax = max{oij [0]} + rij , xmin = max{oij [0]} − rij ymax = max{oij [1]} + rij , ymin = min{oij [1]} − rij (3)

185

x

∆θmax (ttc)

x

δmax

y y δmid

Figure 5: Personal space of group formations. Each colour represents a different group member. The red disc denotes the reference point of the formation.

tcmin

Step 4. In the final step, we select the new velocity Vnew and formation F new of the group. First, for each candidate formation F cand we determine an optimal solution velocity among its set of admissible velocities AVF cand . In particular, we discretize the set AVF cand into a number of candidate velocities and retain the velocity that minimizes a specific cost function. Similar to [van den Berg et al. 2008], our cost function depends on the deviation from the group’s desired velocity Vdes and the minimal predicted time to collision ttc between the formation’s personal space and the neighbouring agents/obstacles determined in the previous step of our model. An additional cost term g(F cand ) is also included to indicate the penalty of selecting the formation F cand . The cost for a candidate velocity Vcand can now be computed as:

Step 3. In the third step, we determine the admissible velocity domain AVF cand of each candidate formation. In particular, we compute the first N agents that are on collision course with F cand given a certain anticipation time tα . We assume that a collision at some time ttc ≥ 0 occurs when an agent steps into or touches the personal space of the formation. Consequently, collision prediction can be achieved by performing a swept test [Ericson 2005] between an aabb (formation’s personal space) and a disc (agent A), where the motion of the box is determined by the group’s desired velocity and the motion of the disc by the current velocity of the agent. Note that the collision test is performed in the local coordinate frame defined by the candidate formation. In a similar way, we determine the set of threatening static obstacle neighbours of F cand . Then, the set of admissible velocities AVF cand varies according to the minimal time-to-collision ttc between the candidate formation and its neighbouring agents/obstacles; collisions in the far future lead to a rather limited domain, whereas shorter collision times allow larger variation in the admissible velocities to successfully resolve imminent collisions. The relationship between the minimal timeto-collision and the set of admissible velocities is mathematically defined based on the motion analysis of pedestrian interactions in controlled scenarios [Karamouzas and Overmars 2010]. Figure 6 plots the maximum angle ∆θmax that the formation can deviate from its desired direction of motion as a function of the minimal ttc for some user-defined tcmin , tcmid and dmid and dmax parameters. Having retrieved the maximum deviation angle ∆θmax , we can determine the admissible orientation domain OF cand of the candidate formation as follows:

T

cost(V

UF cand =

,F

cand

) = κ1

tα − ttc kVcand − Vdes k cand +κ2 +κ3 g(F ), 2U max tα (7)

The term g(F cand ) indicates the deformation energy that is required to deform from the abreast formation into the candidate Abreast cand W −W F

formation and is approximated by the ratio W Abreast −W River , where W i indicates the width of the personal space of formation i, that is the width of its corresponding aabb. As can be observed, the deformation cost becomes zero when the candidate formation is the line-abreast formation. This is in accordance with the empirical observations mentioned in Section 3 indicating that group members prefer to walk side by side since they can easily communicate with each other. In contrast, the deformation cost is maximal when the river-like formation is selected, as there is no social communication between the members.

(4)

des

 U des (1 − e−ttc ), U des +

cand

where the constants κ1 , κ2 , κ3 define the weights of the specific cost terms and can vary to simulate a variety of avoidance behaviours.

where nθ = [cos θ, sin θ] and θ is the orientation angle derived from the group’s desired velocity Vdes . Similarly, the admissible speed domain UF cand is approximated by the following piecewise function:  des U , 

tα

Figure 6: Maximum orientation deviation as a function of the minimal predicted time to collision.

We note that the personal space of the formation is not determined by its minimum bounding box. Instead, our representation focuses on the lateral space that the formation occupies, which allows us to determine how soon the front of the formation will collide with static and dynamic obstacles.

OF cand = { nθ , θ ∈ [θdes − ∆θmax , θdes + ∆θmax ] },

tcmid ttc

Having retrieved the optimal solution velocity for each candidate formation, we retain the velocity Vnew that has the minimal cost among all the solutions velocities. Its corresponding formation is also selected as the new group formation F new for the current step of the simulation:

if tcmin < ttc U max −U des ttc

e



,

(V

otherwise

,F

new

)=

argmin F cand ∈ AF, Vcand ∈ AV

(5)

where U des = Vdes denotes the preferred speed of the group and U max ≥ U des defines the maximum speed at which the group members can move. Then, we can derive the set of feasible velocities AVF cand from the admissible orientation and speed domains: AVF cand = {u nθ | u ∈ UF cand ∧ nθ ∈ OF cand }

new

{cost(V

cand

,F

cand

)}

F cand

(8)

5

Avoidance Maneuvers for the Members

In this section, we elaborate on the second phase of our global approach. The goal here is to plan the motion of each group member Aij based on the new velocity Vnew and formation F new of its group Gi . Our proposed model consists of two steps.

(6)

We refer the reader to [Karamouzas and Overmars 2010] for a more detailed explanation of this step.

186

Vinew

Vinew

Figure 8: Determining the desired direction of motion by left: moving towards the desired future positions, right: finding the best matching between the current and the desired positions of the group members. Figure 7: Determining the desired direction of motion based on left: the new formation, right: the new formation extrapolated in the future. Current and future positions are represented with dark and light colours respectively.

5.1

Using the ndes ij , the agent Aij is now able to find its place in the desired new formation. However, it should also adapt its desired speed udes ij accordingly, so that the entire group Gi can smoothly progress from its current formation to the new one. In other words, the agent Aij has to be able to slow down and wait for the other members or speed up if it is lagging behind. Thus, we first determine for both new the reference point pF and the Aij the distance that they have i to travel from their current positions to their corresponding extrapolated future positions. We then define the difference ddif f between these distances as:

Computing a Desired Velocity

des In the first step, we retrieve the new desired velocity vij of the des member Aij , that is its desired direction of motion nij and speed udes ij . Ideally, at every cycle of the simulation, the agent Aij should move toward its corresponding position in the new formation F new . Note, though, that this can introduce oscillations, e.g. if the agent is already ahead of its desired position or too close (see Figure 7). To alleviate this, we extrapolate the reference point of the new formation into the near future based on the new velocity Vnew , that is: new p0i = pF + Vnew ∗ textrapolate , (9) i

0new

ddif f = kxF ij

N

N

j=1

j=1

new

tmin =

kVnew k

new

k.

(12)

The new desired speed of Aij can now be computed as: new udes k + ddif f /textrapolate ij = kVi

(13)

des des Finally, the agent’s desired velocity is given by vij = udes ij nij .

where to ensure smooth behaviour and avoid the aforementioned oscillations, the parameter textrapolate is clamped to a minimum value tmin determined as: new k + maxkoF maxkxij − pF ij i

− xij k − kp0i − pF i

5.2

Local Collision Avoidance

des In the second step of our approach, the desired velocity vij of the group agent Aij is used as an input to a local collision avoidance method in order to retrieve a collision-free velocity for the agent and update its position. The agent still needs to avoid collisions with the other members of its group, as well as with nearby individuals and groups, due to the fact that in the first phase of our algorithm we plan for the personal space of each candidate group formation and not for the actual formation of the group. As a result, collisions may occur that need to be resolved.

k (10)

It can be proven by contradiction that choosing a textrapolate ≥ tmin guarantees that the desired position of Aij lies always ahead of its current location xij . Based on the extrapolated position of the reference point, we can now deduce the future formation F 0new of the group around that position and also determine the corresponding desired future location of the agent Aij from Equation (1), where the group’s desired Vnew direction of motion is estimated from Vnew , that is n = kV new k . Then, the desired direction of motion of Aij (see Figure 7) can be computed as: 0new xF − xij ij ndes (11) ij = F 0new kxij − xij k

In general, any local method that is based on collision prediction can be used (see e.g. [Karamouzas et al. 2009; van den Berg et al. 2008]). In our implementation, we used the velocity-based approach proposed in [Karamouzas and Overmars 2010] since it is elaborated from experimental interactions data resulting in smooth and collision-free motions. Note that, in the local method, the agent Aij perceives and avoids other groups as single entities, just like individuals do in real-life. Consequently, when solving interactions with outgroup members, the agent takes into account the space that their group occupies, instead of trying to avoid each group member individually. Only when the distance between the group members becomes too large, the group is not considered anymore as one entity but as a collection of individual pedestrians [Cheyne and Efran 1972].

It must be pointed out that the technique described so far assumes 0new that the agents respect their relative formation positions oF ij while selecting their new direction of motion. However, this may not always lead to a desired behaviour, as the group members may need to put a lot of effort and perform a number of extra avoidance maneuvers to safely reach their desired (future) locations without colliding with each other (see Figure 8, left). Thus, an alternative approach, is to find a one-one correspondence between the current positions of the group members and the positions in the extrapolated future formation F 0new while minimizing the total distance that the members have to travel (Figure 8, right). In all of our simulations, this approach was used to derive the new desired direction of notion ndes ij of each agent Aij resulting in a smooth transitioning of the group formation.

6

Results

We have implemented our approach to validate the quality of the generated motions and test its applicability in real-time applications. All experiments were performed on a 2.4 GHz Core 2 Duo CPU (on a single thread) with an ATI Radeon HD 4800 GPU . At runtime, the desired velocity of each agent group was provided by the Corridor Map Method [Kamphuis and Overmars 2004].

187

(a) Crowd simulation in a shopping mall

(b) Another view of the simulation

(c) A still from a video of the mall

Figure 9: Group interactions at the Hoog Catharijne shopping mall in Utrecht, Netherlands (right). Our method is able to predict the emergence of empirically observed walking patterns generating convincing motions (left, middle).

6.1

Quality Evaluation

haviour of a group. The first metric measures the distortion of the group, that is how much the actual formation of a group Gi deviates from its desired abreast formation F0 . For a given time step of the simulation, the distortion Di of Gi is defined as:

We demonstrate the capability of our approach in different scenarios. The resulting simulations can be seen in the video accompanying this paper.

Di =

Deformable Formations: One of the key concepts of our approach is the ability of the groups to dynamically adapt their formations in confined and dynamic environments in order to safely navigate towards their goals. At the same time, the groups favour formations that facilitate the communication between their members, exhibiting behaviour similar to the one observed in real pedestrian groups. For example, we simulate a group of three agents that have to navigate through a narrow corridor (Figure 1(d)). Using our approach, the group adapts a V-shape formation, allowing its members to avoid collisions and communicate with each other. In contrast, using only a local collision avoidance method results in a wedge (inverse V-like) formation. Although such formation is more efficient than the V-shape, it is not suited for social interactions between the group members, since the leader has to turn his back on the other two members. Another example is shown in Figure 1(a), where a group of two agents has to pass through a doorway. Since only one agent can fit through the door, the group gradually adapts a river-like formation ensuring a collision-free motion.

N X

0 kxij − xF ij k,

(14)

j=1

The second metric is the longitudinal dispersion of the group, measured by the distance between the front sfi and the back sbi of the group Gi . The front of a group is determined by the member that is leading the group, that is the agent whose position in the local coordinate system centered and oriented on the centroid of the group has the maximum x coordinate. Similarly, the back of the group is determined by the minimum x local coordinate among the group members. Using our proposed evaluation metrics we evaluated the quality of the shopping mall simulation. The results showed that, on average, only 2.3% ± 0.2 of the time a group adapts a formation different than the line-abreast and the V-like formation. This demonstrates the ability of the group members to find comfortable walking positions supporting the communication with each other. Additionally, the average dispersion and distortion of the groups are quite low, 0.45 ± 0.12, 0.16 ± 0.3 respectively, indicating that the groups remain as gathered as possible during their navigation and deform only if it is necessary. A near-future objective is to compare the numerical simulations with empirical data of pedestrian groups.

Interaction Scenarios: Figure 1(b) shows a group-group interaction in a narrow corridor. As can be observed in the companion video both groups have to change their formations in order to safely reach their goals. However, when a group has enough space to maneuver, it prefers to slightly deviate from its desired direction and maintain its abreast formation, rather than adapt its configuration. Another interaction example is depicted in Figure 1(c), in which a fast moving group encounters a slower moving group heading into the same direction. Using our approach, the faster group tries to stay as coherent as possible by adapting its formation for a short time period before getting back to a line formation. In contrast, planning separately for each group member results in the faster moving group to split in order to avoid the slower moving agents.

6.2

Performance

Obviously the running time of our approach is significantly affected by the number of candidate formations F cand that we evaluate in the first phase of our algorithm during each cycle of the simulation. As mentioned in Section 4, these formations are derived by linearly blending between the current formation of a group and its k = N + 2 desired formations. We have experimented with different test-case scenarios in a number of virtual environments and found out that by considering three intermediate formations between the current formation and each desired one, that is in total k ∗ 3 + k + 1 formations, our method generates smooth motions without imposing any significant overhead to the CPU usage.

Shopping Mall: Figure 9 shows the simulation of a crowd of 300 agents entering/exiting a shopping mall area. The crowd consists of pairs, triples and individual agents. As can be inferred from the supplementary video, our model is in accordance with empirical data of pedestrian groups collected by means of video recordings.

To test the performance of our approach, we selected a varying number of groups consisting of 2-3 members and placed them randomly across an environment (100×100m) void of obstacles. Each group had to advance toward a random goal position avoiding collisions with the other moving groups; when it had reached its destination, a new goal was chosen. Table 1 highlights the performance of our approach. As can be inferred from the table, the running time of our algorithm scales almost linearly with the number of simulated groups. Even for 500 groups we were able to generate more than

Besides the visual inspection of the generated simulations, we are also interested in a quantitative evaluation of our model. As a result, we have devised a number of simple quantitative metrics that can capture the overall steering behaviour of the simulated groups. In particular, we compute the percentage of the time that a group is not in its line-abreast or V-shape formation. Additionally, since we aim for groups of agents that remain as coherent as possible, we propose two additional metrics to determine the coherent be-

188

# Groups

# Agents

100 200 300 400 500

257 507 767 1013 1267

Avg Running Time (msec/frames) 5.25 18.25 32.44 52.98 77.22

cost terms of our evaluation function (Equation (7)). A high penalty for the deviation or time to collision cost term will force the group to adapt a river-like formation, whereas a relatively high penalty for the deformation cost will make the group stick to its desired abreast formation. In our simulations, we set the default values to κ1 = 1.0, κ2 = 0.5, κ3 = 0.2. In all of our test cases, these parameters generated the best agreement with the empirical observations, leading at the same time to visually convincing simulations.

Table 1: Performance of our approach for a varying number of small groups. The reported times are simulation only.

Another approach to model pedestrian groups is the continuum crowd formulation [Treuille et al. 2006] which unifies global planning and local collision avoidance into a single framework. Although this method exhibits emergent phenomena observed in real crowds, it is mainly suited for the simulation of a limited number of large homogeneous groups and is prohibitively computationally expensive for planning the movements of a large number of small groups that have distinct characteristics and goals. Finally, Kwon et al. [2008] have recently proposed a spectral-based approach for synthesizing realistic group motions. A similar idea was also employed in [Takahashi et al. 2009] in order to control the spatial arrangements of groups of multiple individuals in a number of different applications. However, both approaches aim to govern the macroscopic behaviour of agent groups, whereas our goal is to simulate group interactions at the microscopic level. Additionally, the running times of both techniques are too high for real-time applications such as interactive virtual worlds.

10f ps in our benchmark scenario. Since our current implementation is unoptimized and uses only one CPU thread for computing the motions of the agents, it is clear that our approach can be used for real-time simulation of small groups of characters.

7

Discussion

In this section, we compare our approach with prior work and also discuss some of the limitations of our method. Comparisons: The most common approach to simulate group movement is the flocking technique introduced by Reynolds [1987]. Flocking generates convincing motions, but is mainly applicable to model herds, flocks or schools. Later, Reynolds extended the flocking model by incorporating additional steering behaviours such as the leader-follower behaviour [Reynolds 1999]. Based on his work, a considerable amount of research has emerged to model virtual crowds using simple local rules (see e.g [Loscos et al. 2003; Pelechano et al. 2007]). However, all these approaches are less suited for simulating small groups of virtual humans, since they do not take into account the way that friends or couples walk next to each other and how groups are perceived and avoided by other groups and walking individuals.

Limitations: Like any other model, our approach has some limitations. Although our algorithm can deal with dynamic and challenging scenarios by adapting the group formation, the final motions of the group members are dependent on the local avoidance mechanism of the second phase of our algorithm (see Section 5.2). Thus, since no local method can absolutely guarantee collision-free motion, some collisions may occur in rather complex environments. It should also be noted that our method only aims to control the local interactions of small groups. As a result, a global path planning approach should be used to guide the global motions of the groups such that they cannot get trapped in local minima due to the presence of obstacles (see e.g. [Lamarche and Donikian 2004; Geraerts and Overmars 2007]). Finally, it must be pointed out that our method is not intended for simulating groups of characters in densely packed scenarios. We believe that in these scenarios the focus is on the macroscopic rather than on the microscopic simulation of the crowd, and thus, approaches based on continuum dynamics should be used [Hughes 2003; Treuille et al. 2006].

More recently, Singh et al. [2010] proposed an intuitive method that enables agents to reach their destinations while maintaining a specific geometric formation. In their approach, safety plays an important role in the stability of the group formations. The requirements, though, for controlling a small group of friends are not the same. In such groups, the communication between the group members significantly affects the configuration of the group. To that end, our current research is closely related to the recent work of Moussa¨ıd et al. [2010], since it is elaborated from their empirical observations regarding the spatial organization of pedestrian groups. Using these observations, Moussa¨ıd and his colleagues extended Helbing’s social force model to account for small groups. Their approach generates macroscopically plausible behaviour. However, it lacks anticipation and prediction. Therefore, the groups interact when they get sufficiently close resulting in unnatural motions and oscillatory behaviour. The problem becomes more obvious in large and cluttered environments, with group members constantly changing their orientations, pushing each other and moving back and forth while interacting with other groups and static obstacles. In contrast, our approach uses the velocity space to plan the avoidance maneuvers of the groups. This ensures smooth and oscillation-free motions, generating at the same time the correct microscopic (local) behaviour for the pedestrian groups.

8

Conclusions and Future Work

We presented a novel method for simulating the local motions of small groups of virtual humans. The intuition behind our approach is based on observed behaviour of real crowds; in real-life couples, friends or families tend to walk next to each other forming groups of two or three members. Similarly, our model considers pairs and triples of characters and uses a two-step algorithm to ensure that the groups will stay as coherent as possible while avoiding collisions with other groups, individuals and static obstacles. We have demonstrated the potential of our method through a wide range of test-case scenarios. Our method is able to predict the emergence of empirically observed walking patterns generating smooth and visually convincing motions. A number of metrics were also proposed to quantitatively evaluate the quality of our model. Currently, we are developing a tool to semi-automatically track pedestrians in outdoor environments. This will allow us to gain more insight into the avoidance behaviour of pairs and small groups of pedestrians and extend our model accordingly. Joining and splitting of pedestrian groups, a common event in virtual environments, is also one of the research topics currently under investigation.

Peters and Ennis [2009] have also recently proposed a model to simulate plausible behaviours of small groups. Similar to our approach, their method takes into account the different spatial patterns observed in real pedestrian groups. However, their approach is rulebased often requiring careful parameter tuning to obtain a desired simulation result. In contrast, our model automatically solves interactions between groups without the use of any explicit rules. The behaviour of each group depends only on the weights of the three

189

Acknowledgments

L AVALLE , S. 2006. Planning algorithms. Cambridge University Press.

The authors would like to thank Peter Heil for his help in developing an early version of this work. This research has been supported by the GATE project, funded by the Netherlands Organization for Scientific Research (NWO) and the Netherlands ICT Research and Innovation Authority (ICT Regie).

L EE , K., C HOI , M., H ONG , Q., AND L EE , J. 2007. Group behavior from video: a data-driven approach to crowd simulation. In SCA ’07: ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 109–118.

References

L ERNER , A., C HRYSANTHOU , Y., AND L ISCHINSKI , D. 2007. Crowds by example. Computer Graphics Forum 26, 655–664. L I , T., AND C HOU , H. 2003. Motion planning for a crowd of robots. In IEEE International Conference on Robotics and Automation, ICRA’03, vol. 3.

BAYAZIT, O. B., L IEN , J.-M., AND A MATO , N. M. 2003. Better group behaviors in complex environments using global roadmaps. In ICAL 2003: Proceedings of the eighth international conference on Artificial life, 362–370.

L OSCOS , C., M ARCHAL , D., AND M EYER , A. 2003. Intuitive crowd behaviour in dense urban environments using local laws. Theory and Practice of Computer Graphics.

B RAUN , A., M USSE , S. R., O LIVEIRA , L. P. L. D ., AND B OD MANN , B. E. J. 2003. Modeling individual behaviors in crowd simulation. In CASA ’03: Proceedings of the 16th International Conference on Computer Animation and Social Agents, 143.

M OUSSA¨I D , M., P EROZO , N., G ARNIER , S., H ELBING , D., AND T HERAULAZ , G. 2010. The walking behaviour of pedestrian social groups and its impact on crowd dynamics. PLoS ONE 5, 4 (April), e10047.

B ROGAN , D. C., AND H ODGINS , J. K. 1997. Group behaviors for systems with significant dynamics. Autonomous Robots 4, 137–153.

M USSE , S. R., AND T HALMANN , D. 1997. A model of human crowd behavior: Group inter-relationship and collision detection analysis. In Computer Animation and Simulation ’97, 39–51.

C HEYNE , J., AND E FRAN , M. 1972. The effect of spatial and interpersonal variables on the invasion of group controlled territories. Sociometry 35, 3, 477–489.

PARIS , S., P ETTR E´ , J., AND D ONIKIAN , S. 2007. Pedestrian reactive navigation for crowd simulation: a predictive approach. Computer Graphics Forum 26, 3, 665–674.

C OLEMAN , J., AND JAMES , J. 1961. The equilibrium size distribution of freely-forming groups. Sociometry 24, 1, 36–45. E RICSON , C. 2005. Real-time collision detection. Morgan Kaufmann.

P ELECHANO , N., A LLBECK , J. M., AND BADLER , N. I. 2007. Controlling individual agents in high-density crowd simulation. In SCA ’07: Symposium on Computer Animation, 99–108.

F UNGE , J., T U , X., AND T ERZOPOULOS , D. 1999. Cognitive modeling: knowledge, reasoning and planning for intelligent characters. In Proceedings of SIGGRAPH ’99, 29–38.

P ETERS , C., AND E NNIS , C. 2009. Modeling groups of plausible virtual pedestrians. IEEE Comput. Graph. Appl. 29, 4, 54–63.

G ERAERTS , R., AND OVERMARS , M. 2007. The corridor map method: A general framework for real-time high-quality path planning. Computer Animation and Virtual Worlds 18, 107–119.

P ETTR E´ , J., O NDREJ , J., O LIVIER , A.-H., C R E´ TUAL , A., AND D ONIKIAN , S. 2009. Experiment-based modeling, simulation and validation of interactions between virtual walkers. In SCA ’09: Symposium on Computer Animation, 189–198.

G UY, S., ET AL . 2009. Clearpath: highly parallel collision avoidance for multi-agent simulation. In SCA ’09: Symposium on Computer Animation, 177–187.

Q IU , F., H U , X., WANG , X., AND K ARMAKAR , S. 2008. Modeling social group structures in pedestrian crowds. In 7th Int. Conf. on System Simulation and Scientific Computing.

H ELBING , D., FARKAS , I., AND V ICSEK , T. 2000. Simulating dynamical features of escape panic. Nature 407, 6803, 487–490.

R EYNOLDS , C. W. 1987. Flocks, herds, and schools: A distributed behavioral model. Computer Graphics 21, 4, 24–34.

H UGHES , R. 2003. The flow of human crowds. Annu. Rev. Fluid Mech 35, 169–182.

R EYNOLDS , C. W. 1999. Steering behaviors for autonomous characters. In Proc. of Game Developers Conference, 763–782.

K AMPHUIS , A., AND OVERMARS , M. 2004. Finding paths for coherent groups using clearance. In Proc. Eurographics/ACM SIGGRAPH Symposium on Computer Animation, 19–28.

S HAO , W., AND T ERZOPOULOS , D. 2007. Autonomous pedestrians. Graphical Models 69, 5-6, 246–274. S INGH , S., K APADIA , M., FALOUTSOS , P., AND S CHUERMAN , M. 2010. Situation agents: agent-based externalized steering logic. Computer Animation and Virtual Worlds 21, 3-4, 267– 276.

K ARAMOUZAS , I., AND OVERMARS , M. 2010. Simulating human collision avoidance using a velocity-based approach. In VRIPHYS ’10: The 7th Workshop on Virtual Reality Interaction and Physical Simulation.

TAKAHASHI , S., YOSHIDA , K., K WON , T., L EE , K. H., L EE , J., AND S HIN , S. Y. 2009. Spectral-based group formation control. Computer Graphics Forum 28, 2, 639–648.

K ARAMOUZAS , I., H EIL , P., VAN B EEK , P., AND OVERMARS , M. H. 2009. A predictive collision avoidance model for pedestrian simulation. In Proceedings of Motion in Games, 41–52.

T REUILLE , A., C OOPER , S., AND P OPOVI C´ , Z. 2006. Continuum crowds. ACM Transactions on Graphics 25, 3, 1160–1168.

K WON , T., L EE , K. H., L EE , J., AND TAKAHASHI , S. 2008. Group motion editing. ACM Trans. Graph. 27, 3, 1–8.

B ERG , J. P., L IN , M., AND M ANOCHA , D. 2008. Reciprocal velocity obstacles for real-time multi-agent navigation. In IEEE Conference on Robotics and Automation, 1928–1935.

VAN DEN

L AMARCHE , F., AND D ONIKIAN , S. 2004. Crowd of virtual humans: a new approach for real time navigation in complex and structured environments. Computer Graphics Forum 23, 509– 518.

190